internal/ceres/conjugate_gradients_solver.h - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2023 Google Inc. All rights reserved.
 // http://ceres-solver.org/
 //
 // Redistribution and use in source and binary forms, with or without
 // modification, are permitted provided that the following conditions are met:
 //
 // * Redistributions of source code must retain the above copyright notice,
 //   this list of conditions and the following disclaimer.
 // * Redistributions in binary form must reproduce the above copyright notice,
 //   this list of conditions and the following disclaimer in the documentation
 //   and/or other materials provided with the distribution.
 // * Neither the name of Google Inc. nor the names of its contributors may be
 //   used to endorse or promote products derived from this software without
 //   specific prior written permission.
 //
 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
 // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 // POSSIBILITY OF SUCH DAMAGE.
 //
 // Author: sameeragarwal@google.com (Sameer Agarwal)
 //
 // Preconditioned Conjugate Gradients based solver for positive
 // semidefinite linear systems.

 #ifndef CERES_INTERNAL_CONJUGATE_GRADIENTS_SOLVER_H_
 #define CERES_INTERNAL_CONJUGATE_GRADIENTS_SOLVER_H_

 #include <cmath>
 #include <cstddef>
 #include <utility>

 #include "ceres/eigen_vector_ops.h"
 #include "ceres/internal/disable_warnings.h"
 #include "ceres/internal/eigen.h"
 #include "ceres/internal/export.h"
 #include "ceres/linear_operator.h"
 #include "ceres/linear_solver.h"
 #include "ceres/stringprintf.h"
 #include "ceres/types.h"
 #include "glog/logging.h"

 namespace ceres::internal {

 // Interface for the linear operator used by ConjugateGradientsSolver.
 template <typename DenseVectorType>
 class ConjugateGradientsLinearOperator {
  public:
   ~ConjugateGradientsLinearOperator() = default;
   virtual void RightMultiplyAndAccumulate(const DenseVectorType& x,
                                           DenseVectorType& y) = 0;
 };

 // Adapter class that makes LinearOperator appear like an instance of
 // ConjugateGradientsLinearOperator.
 class LinearOperatorAdapter : public ConjugateGradientsLinearOperator<Vector> {
  public:
   LinearOperatorAdapter(LinearOperator& linear_operator)
       : linear_operator_(linear_operator) {}

   void RightMultiplyAndAccumulate(const Vector& x, Vector& y) final {
     linear_operator_.RightMultiplyAndAccumulate(x, y);
   }

  private:
   LinearOperator& linear_operator_;
 };

 // Options to control the ConjugateGradientsSolver. For detailed documentation
 // for each of these options see linear_solver.h
 struct ConjugateGradientsSolverOptions {
   int min_num_iterations = 1;
   int max_num_iterations = 1;
   int residual_reset_period = 10;
   double r_tolerance = 0.0;
   double q_tolerance = 0.0;
   ContextImpl* context = nullptr;
   int num_threads = 1;
 };

 // This function implements the now classical Conjugate Gradients algorithm of
 // Hestenes & Stiefel for solving positive semidefinite linear systems.
 // Optionally it can use a preconditioner also to reduce the condition number of
 // the linear system and improve the convergence rate. Modern references for
 // Conjugate Gradients are the books by Yousef Saad and Trefethen & Bau. This
 // implementation of CG has been augmented with additional termination tests
 // that are needed for forcing early termination when used as part of an inexact
 // Newton solver.
 //
 // This implementation is templated over DenseVectorType and then in turn on
 // ConjugateGradientsLinearOperator, which allows us to write an abstract
 // implementaion of the Conjugate Gradients algorithm without worrying about how
 // these objects are implemented or where they are stored. In particular it
 // allows us to have a single implementation that works on CPU and GPU based
 // matrices and vectors.
 //
 // scratch must contain pointers to four DenseVector objects of the same size as
 // rhs and solution. By asking the user for scratch space, we guarantee that we
 // will not perform any allocations inside this function.
 template <typename DenseVectorType>
 LinearSolver::Summary ConjugateGradientsSolver(
     const ConjugateGradientsSolverOptions options,
     ConjugateGradientsLinearOperator<DenseVectorType>& lhs,
     const DenseVectorType& rhs,
     ConjugateGradientsLinearOperator<DenseVectorType>& preconditioner,
     DenseVectorType* scratch[4],
     DenseVectorType& solution) {
   auto IsZeroOrInfinity = [](double x) {
     return ((x == 0.0) || std::isinf(x));
   };

   DenseVectorType& p = *scratch[0];
   DenseVectorType& r = *scratch[1];
   DenseVectorType& z = *scratch[2];
   DenseVectorType& tmp = *scratch[3];

   LinearSolver::Summary summary;
   summary.termination_type = LinearSolverTerminationType::NO_CONVERGENCE;
   summary.message = "Maximum number of iterations reached.";
   summary.num_iterations = 0;

   const double norm_rhs = Norm(rhs, options.context, options.num_threads);
   if (norm_rhs == 0.0) {
     SetZero(solution, options.context, options.num_threads);
     summary.termination_type = LinearSolverTerminationType::SUCCESS;
     summary.message = "Convergence. |b| = 0.";
     return summary;
   }

   const double tol_r = options.r_tolerance * norm_rhs;

   SetZero(tmp, options.context, options.num_threads);
   lhs.RightMultiplyAndAccumulate(solution, tmp);

   // r = rhs - tmp
   Axpby(1.0, rhs, -1.0, tmp, r, options.context, options.num_threads);

   double norm_r = Norm(r, options.context, options.num_threads);
   if (options.min_num_iterations == 0 && norm_r <= tol_r) {
     summary.termination_type = LinearSolverTerminationType::SUCCESS;
     summary.message =
         StringPrintf("Convergence. |r| = %e <= %e.", norm_r, tol_r);
     return summary;
   }

   double rho = 1.0;

   // Initial value of the quadratic model Q = x'Ax - 2 * b'x.
   // double Q0 = -1.0 * solution.dot(rhs + r);
   Axpby(1.0, rhs, 1.0, r, tmp, options.context, options.num_threads);
   double Q0 = -Dot(solution, tmp, options.context, options.num_threads);

   for (summary.num_iterations = 1;; ++summary.num_iterations) {
     SetZero(z, options.context, options.num_threads);
     preconditioner.RightMultiplyAndAccumulate(r, z);

     const double last_rho = rho;
     // rho = r.dot(z);
     rho = Dot(r, z, options.context, options.num_threads);
     if (IsZeroOrInfinity(rho)) {
       summary.termination_type = LinearSolverTerminationType::FAILURE;
       summary.message = StringPrintf("Numerical failure. rho = r'z = %e.", rho);
       break;
     }

     if (summary.num_iterations == 1) {
       Copy(z, p, options.context, options.num_threads);
     } else {
       const double beta = rho / last_rho;
       if (IsZeroOrInfinity(beta)) {
         summary.termination_type = LinearSolverTerminationType::FAILURE;
         summary.message = StringPrintf(
             "Numerical failure. beta = rho_n / rho_{n-1} = %e, "
             "rho_n = %e, rho_{n-1} = %e",
             beta,
             rho,
             last_rho);
         break;
       }
       // p = z + beta * p;
       Axpby(1.0, z, beta, p, p, options.context, options.num_threads);
     }

     DenseVectorType& q = z;
     SetZero(q, options.context, options.num_threads);
     lhs.RightMultiplyAndAccumulate(p, q);
     const double pq = Dot(p, q, options.context, options.num_threads);
     if ((pq <= 0) || std::isinf(pq)) {
       summary.termination_type = LinearSolverTerminationType::NO_CONVERGENCE;
       summary.message = StringPrintf(
           "Matrix is indefinite, no more progress can be made. "
           "p'q = %e. |p| = %e, |q| = %e",
           pq,
           Norm(p, options.context, options.num_threads),
           Norm(q, options.context, options.num_threads));
       break;
     }

     const double alpha = rho / pq;
     if (std::isinf(alpha)) {
       summary.termination_type = LinearSolverTerminationType::FAILURE;
       summary.message = StringPrintf(
           "Numerical failure. alpha = rho / pq = %e, rho = %e, pq = %e.",
           alpha,
           rho,
           pq);
       break;
     }

     // solution = solution + alpha * p;
     Axpby(1.0,
           solution,
           alpha,
           p,
           solution,
           options.context,
           options.num_threads);

     // Ideally we would just use the update r = r - alpha*q to keep
     // track of the residual vector. However this estimate tends to
     // drift over time due to round off errors. Thus every
     // residual_reset_period iterations, we calculate the residual as
     // r = b - Ax. We do not do this every iteration because this
     // requires an additional matrix vector multiply which would
     // double the complexity of the CG algorithm.
     if (summary.num_iterations % options.residual_reset_period == 0) {
       SetZero(tmp, options.context, options.num_threads);
       lhs.RightMultiplyAndAccumulate(solution, tmp);
       Axpby(1.0, rhs, -1.0, tmp, r, options.context, options.num_threads);
       // r = rhs - tmp;
     } else {
       Axpby(1.0, r, -alpha, q, r, options.context, options.num_threads);
       // r = r - alpha * q;
     }

     // Quadratic model based termination.
     //   Q1 = x'Ax - 2 * b' x.
     // const double Q1 = -1.0 * solution.dot(rhs + r);
     Axpby(1.0, rhs, 1.0, r, tmp, options.context, options.num_threads);
     const double Q1 = -Dot(solution, tmp, options.context, options.num_threads);

     // For PSD matrices A, let
     //
     //   Q(x) = x'Ax - 2b'x
     //
     // be the cost of the quadratic function defined by A and b. Then,
     // the solver terminates at iteration i if
     //
     //   i * (Q(x_i) - Q(x_i-1)) / Q(x_i) < q_tolerance.
     //
     // This termination criterion is more useful when using CG to
     // solve the Newton step. This particular convergence test comes
     // from Stephen Nash's work on truncated Newton
     // methods. References:
     //
     //   1. Stephen G. Nash & Ariela Sofer, Assessing A Search
     //   Direction Within A Truncated Newton Method, Operation
     //   Research Letters 9(1990) 219-221.
     //
     //   2. Stephen G. Nash, A Survey of Truncated Newton Methods,
     //   Journal of Computational and Applied Mathematics,
     //   124(1-2), 45-59, 2000.
     //
     const double zeta = summary.num_iterations * (Q1 - Q0) / Q1;
     if (zeta < options.q_tolerance &&
         summary.num_iterations >= options.min_num_iterations) {
       summary.termination_type = LinearSolverTerminationType::SUCCESS;
       summary.message =
           StringPrintf("Iteration: %d Convergence: zeta = %e < %e. |r| = %e",
                        summary.num_iterations,
                        zeta,
                        options.q_tolerance,
                        Norm(r, options.context, options.num_threads));
       break;
     }
     Q0 = Q1;

     // Residual based termination.
     norm_r = Norm(r, options.context, options.num_threads);
     if (norm_r <= tol_r &&
         summary.num_iterations >= options.min_num_iterations) {
       summary.termination_type = LinearSolverTerminationType::SUCCESS;
       summary.message =
           StringPrintf("Iteration: %d Convergence. |r| = %e <= %e.",
                        summary.num_iterations,
                        norm_r,
                        tol_r);
       break;
     }

     if (summary.num_iterations >= options.max_num_iterations) {
       break;
     }
   }

   return summary;
 }

 }  // namespace ceres::internal

 #include "ceres/internal/reenable_warnings.h"

 #endif  // CERES_INTERNAL_CONJUGATE_GRADIENTS_SOLVER_H_
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2023 Google Inc. All rights reserved.
	// http://ceres-solver.org/
	//
	// Redistribution and use in source and binary forms, with or without
	// modification, are permitted provided that the following conditions are met:
	//
	// * Redistributions of source code must retain the above copyright notice,
	// this list of conditions and the following disclaimer.
	// * Redistributions in binary form must reproduce the above copyright notice,
	// this list of conditions and the following disclaimer in the documentation
	// and/or other materials provided with the distribution.
	// * Neither the name of Google Inc. nor the names of its contributors may be
	// used to endorse or promote products derived from this software without
	// specific prior written permission.
	//
	// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
	// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
	// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
	// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
	// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
	// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
	// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
	// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
	// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
	// POSSIBILITY OF SUCH DAMAGE.
	//
	// Author: sameeragarwal@google.com (Sameer Agarwal)
	//
	// Preconditioned Conjugate Gradients based solver for positive
	// semidefinite linear systems.

	#ifndef CERES_INTERNAL_CONJUGATE_GRADIENTS_SOLVER_H_
	#define CERES_INTERNAL_CONJUGATE_GRADIENTS_SOLVER_H_

	#include <cmath>
	#include <cstddef>
	#include <utility>

	#include "ceres/eigen_vector_ops.h"
	#include "ceres/internal/disable_warnings.h"
	#include "ceres/internal/eigen.h"
	#include "ceres/internal/export.h"
	#include "ceres/linear_operator.h"
	#include "ceres/linear_solver.h"
	#include "ceres/stringprintf.h"
	#include "ceres/types.h"
	#include "glog/logging.h"

	namespace ceres::internal {

	// Interface for the linear operator used by ConjugateGradientsSolver.
	template <typename DenseVectorType>
	class ConjugateGradientsLinearOperator {
	public:
	~ConjugateGradientsLinearOperator() = default;
	virtual void RightMultiplyAndAccumulate(const DenseVectorType& x,
	DenseVectorType& y) = 0;
	};

	// Adapter class that makes LinearOperator appear like an instance of
	// ConjugateGradientsLinearOperator.
	class LinearOperatorAdapter : public ConjugateGradientsLinearOperator<Vector> {
	public:
	LinearOperatorAdapter(LinearOperator& linear_operator)
	: linear_operator_(linear_operator) {}

	void RightMultiplyAndAccumulate(const Vector& x, Vector& y) final {
	linear_operator_.RightMultiplyAndAccumulate(x, y);
	}

	private:
	LinearOperator& linear_operator_;
	};

	// Options to control the ConjugateGradientsSolver. For detailed documentation
	// for each of these options see linear_solver.h
	struct ConjugateGradientsSolverOptions {
	int min_num_iterations = 1;
	int max_num_iterations = 1;
	int residual_reset_period = 10;
	double r_tolerance = 0.0;
	double q_tolerance = 0.0;
	ContextImpl* context = nullptr;
	int num_threads = 1;
	};

	// This function implements the now classical Conjugate Gradients algorithm of
	// Hestenes & Stiefel for solving positive semidefinite linear systems.
	// Optionally it can use a preconditioner also to reduce the condition number of
	// the linear system and improve the convergence rate. Modern references for
	// Conjugate Gradients are the books by Yousef Saad and Trefethen & Bau. This
	// implementation of CG has been augmented with additional termination tests
	// that are needed for forcing early termination when used as part of an inexact
	// Newton solver.
	//
	// This implementation is templated over DenseVectorType and then in turn on
	// ConjugateGradientsLinearOperator, which allows us to write an abstract
	// implementaion of the Conjugate Gradients algorithm without worrying about how
	// these objects are implemented or where they are stored. In particular it
	// allows us to have a single implementation that works on CPU and GPU based
	// matrices and vectors.
	//
	// scratch must contain pointers to four DenseVector objects of the same size as
	// rhs and solution. By asking the user for scratch space, we guarantee that we
	// will not perform any allocations inside this function.
	template <typename DenseVectorType>
	LinearSolver::Summary ConjugateGradientsSolver(
	const ConjugateGradientsSolverOptions options,
	ConjugateGradientsLinearOperator<DenseVectorType>& lhs,
	const DenseVectorType& rhs,
	ConjugateGradientsLinearOperator<DenseVectorType>& preconditioner,
	DenseVectorType* scratch[4],
	DenseVectorType& solution) {
	auto IsZeroOrInfinity = [](double x) {
	return ((x == 0.0) \|\| std::isinf(x));
	};

	DenseVectorType& p = *scratch[0];
	DenseVectorType& r = *scratch[1];
	DenseVectorType& z = *scratch[2];
	DenseVectorType& tmp = *scratch[3];

	LinearSolver::Summary summary;
	summary.termination_type = LinearSolverTerminationType::NO_CONVERGENCE;
	summary.message = "Maximum number of iterations reached.";
	summary.num_iterations = 0;

	const double norm_rhs = Norm(rhs, options.context, options.num_threads);
	if (norm_rhs == 0.0) {
	SetZero(solution, options.context, options.num_threads);
	summary.termination_type = LinearSolverTerminationType::SUCCESS;
	summary.message = "Convergence. \|b\| = 0.";
	return summary;
	}

	const double tol_r = options.r_tolerance * norm_rhs;

	SetZero(tmp, options.context, options.num_threads);
	lhs.RightMultiplyAndAccumulate(solution, tmp);

	// r = rhs - tmp
	Axpby(1.0, rhs, -1.0, tmp, r, options.context, options.num_threads);

	double norm_r = Norm(r, options.context, options.num_threads);
	if (options.min_num_iterations == 0 && norm_r <= tol_r) {
	summary.termination_type = LinearSolverTerminationType::SUCCESS;
	summary.message =
	StringPrintf("Convergence. \|r\| = %e <= %e.", norm_r, tol_r);
	return summary;
	}

	double rho = 1.0;

	// Initial value of the quadratic model Q = x'Ax - 2 * b'x.
	// double Q0 = -1.0 * solution.dot(rhs + r);
	Axpby(1.0, rhs, 1.0, r, tmp, options.context, options.num_threads);
	double Q0 = -Dot(solution, tmp, options.context, options.num_threads);

	for (summary.num_iterations = 1;; ++summary.num_iterations) {
	SetZero(z, options.context, options.num_threads);
	preconditioner.RightMultiplyAndAccumulate(r, z);

	const double last_rho = rho;
	// rho = r.dot(z);
	rho = Dot(r, z, options.context, options.num_threads);
	if (IsZeroOrInfinity(rho)) {
	summary.termination_type = LinearSolverTerminationType::FAILURE;
	summary.message = StringPrintf("Numerical failure. rho = r'z = %e.", rho);
	break;
	}

	if (summary.num_iterations == 1) {
	Copy(z, p, options.context, options.num_threads);
	} else {
	const double beta = rho / last_rho;
	if (IsZeroOrInfinity(beta)) {
	summary.termination_type = LinearSolverTerminationType::FAILURE;
	summary.message = StringPrintf(
	"Numerical failure. beta = rho_n / rho_{n-1} = %e, "
	"rho_n = %e, rho_{n-1} = %e",
	beta,
	rho,
	last_rho);
	break;
	}
	// p = z + beta * p;
	Axpby(1.0, z, beta, p, p, options.context, options.num_threads);
	}

	DenseVectorType& q = z;
	SetZero(q, options.context, options.num_threads);
	lhs.RightMultiplyAndAccumulate(p, q);
	const double pq = Dot(p, q, options.context, options.num_threads);
	if ((pq <= 0) \|\| std::isinf(pq)) {
	summary.termination_type = LinearSolverTerminationType::NO_CONVERGENCE;
	summary.message = StringPrintf(
	"Matrix is indefinite, no more progress can be made. "
	"p'q = %e. \|p\| = %e, \|q\| = %e",
	pq,
	Norm(p, options.context, options.num_threads),
	Norm(q, options.context, options.num_threads));
	break;
	}

	const double alpha = rho / pq;
	if (std::isinf(alpha)) {
	summary.termination_type = LinearSolverTerminationType::FAILURE;
	summary.message = StringPrintf(
	"Numerical failure. alpha = rho / pq = %e, rho = %e, pq = %e.",
	alpha,
	rho,
	pq);
	break;
	}

	// solution = solution + alpha * p;
	Axpby(1.0,
	solution,
	alpha,
	p,
	solution,
	options.context,
	options.num_threads);

	// Ideally we would just use the update r = r - alpha*q to keep
	// track of the residual vector. However this estimate tends to
	// drift over time due to round off errors. Thus every
	// residual_reset_period iterations, we calculate the residual as
	// r = b - Ax. We do not do this every iteration because this
	// requires an additional matrix vector multiply which would
	// double the complexity of the CG algorithm.
	if (summary.num_iterations % options.residual_reset_period == 0) {
	SetZero(tmp, options.context, options.num_threads);
	lhs.RightMultiplyAndAccumulate(solution, tmp);
	Axpby(1.0, rhs, -1.0, tmp, r, options.context, options.num_threads);
	// r = rhs - tmp;
	} else {
	Axpby(1.0, r, -alpha, q, r, options.context, options.num_threads);
	// r = r - alpha * q;
	}

	// Quadratic model based termination.
	// Q1 = x'Ax - 2 * b' x.
	// const double Q1 = -1.0 * solution.dot(rhs + r);
	Axpby(1.0, rhs, 1.0, r, tmp, options.context, options.num_threads);
	const double Q1 = -Dot(solution, tmp, options.context, options.num_threads);

	// For PSD matrices A, let
	//
	// Q(x) = x'Ax - 2b'x
	//
	// be the cost of the quadratic function defined by A and b. Then,
	// the solver terminates at iteration i if
	//
	// i * (Q(x_i) - Q(x_i-1)) / Q(x_i) < q_tolerance.
	//
	// This termination criterion is more useful when using CG to
	// solve the Newton step. This particular convergence test comes
	// from Stephen Nash's work on truncated Newton
	// methods. References:
	//
	// 1. Stephen G. Nash & Ariela Sofer, Assessing A Search
	// Direction Within A Truncated Newton Method, Operation
	// Research Letters 9(1990) 219-221.
	//
	// 2. Stephen G. Nash, A Survey of Truncated Newton Methods,
	// Journal of Computational and Applied Mathematics,
	// 124(1-2), 45-59, 2000.
	//
	const double zeta = summary.num_iterations * (Q1 - Q0) / Q1;
	if (zeta < options.q_tolerance &&
	summary.num_iterations >= options.min_num_iterations) {
	summary.termination_type = LinearSolverTerminationType::SUCCESS;
	summary.message =
	StringPrintf("Iteration: %d Convergence: zeta = %e < %e. \|r\| = %e",
	summary.num_iterations,
	zeta,
	options.q_tolerance,
	Norm(r, options.context, options.num_threads));
	break;
	}
	Q0 = Q1;

	// Residual based termination.
	norm_r = Norm(r, options.context, options.num_threads);
	if (norm_r <= tol_r &&
	summary.num_iterations >= options.min_num_iterations) {
	summary.termination_type = LinearSolverTerminationType::SUCCESS;
	summary.message =
	StringPrintf("Iteration: %d Convergence. \|r\| = %e <= %e.",
	summary.num_iterations,
	norm_r,
	tol_r);
	break;
	}

	if (summary.num_iterations >= options.max_num_iterations) {
	break;
	}
	}

	return summary;
	}

	} // namespace ceres::internal

	#include "ceres/internal/reenable_warnings.h"

	#endif // CERES_INTERNAL_CONJUGATE_GRADIENTS_SOLVER_H_